Talk:Spherical harmonics

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Contents 1 algebraic geometry 2 Table of Harmonics 3 References 4 Different Normalizations 5 Condon-Shortley phase 6 Deleted images 7 To do: add identities 8 Real spherical harmonics 9 Laplace

[edit] algebraic geometry

Edited out this:

'Very closely related to this expansion is the algebraic geometric interpretation of a 2-sphere as the real commutative algebra generated by a,b and c subject to the relation a²+b²+c²-1=0 or the C* interpretation as the C* algebra generated by the self-adjoint elements a,b and c subject to a²+b²+c²-1=0.'

I don't see the close relation. Spherical harmonics do have an upmarket explanation from representation theory; but this isn't (yet) it.

Charles Matthews 12:52, 23 Oct 2003 (UTC)

[edit] Table of Harmonics

Please consider creating a distinct article, possibly Table of spherical harmonics that will hold the list of explicit expressions for l>4. While such a table is not overwhelmingly useful, it can come in handy if one just wants to glance at something and doesn't want to write software to work out some particular case. linas 22:55, 16 August 2005 (UTC)

I had already made the table, just had commented most of it out, so the hardest part was getting it to save without an error message like "Something didn't return a response to your request" error message. Κσυπ Cyp   06:38, 20 August 2005 (UTC)

Yeah, WP has been spitting out that error message recently. I've discovered that quite often, the changes did make it into the database, but the web page didn't get rendered. Thus, if you open a second browser, and load the page, you'll (usually) see that your changes made it. So you don't have to hit the submit button over and over. linas 16:54, 20 August 2005 (UTC)

[edit] References

I added the reference of [Varshalovich, et al.]. This book contains hundred pages of tables and formulas about spherical harmonics and could be very usefull.

Alain Michaud 16:47, 31 Oct 2005 (UTC)

In the angular part of the Laplace equation the index l is shown. It seems to me natural to write down a second formula showing the index m to guide the reader. At the present the index m is quit sudden launched in the notation for the general solution and in the - l<..< m <..<+l inequality.

I might suggest that the azimutal angle name 'fi' will be replaced by 'theta' to conform to the conventions for spatial orientation.

May be the indices l and m could be given here an intuitive geometric interpretation, in advance of the QM interprations for the Schroedinger wave equation of l and m, as integer numbers related to impuls-moment and its projection in the standard ( 'z') -direction.

A last suggestion: compare the Laplace solutions with the solutions of the Schroedinger wave equation for a central potential (like the coulomb field ), to discuss the differences and common aspects for radial - cq angular parts of their solutions.

Hans van der Grift

[edit] Different Normalizations

I have added definitions for the unit power and Schmidt-semi normalized spherical harmonics. Other modifications have been made for consistency and accuracy purposes. I have also added links to software archives that are currently being developed.

Note: We need to mention if the analytic expressions for the first few spherical harmonics use the Condon-Shortley phase or not. Indeed, this should be carefully clarified on the associated Legendre functions link.

Mark Wieczorek, Oct. 26, 2006

[edit] Condon-Shortley phase

"The phase of Y_\ell^{m} necessary for the validity of this complex conjugation relation is the so-called Condon and Shortley phase."

I don't think that I agree with this. It is my understanding the the Condon Shortely phase has nothing to do with the complex conjugation relationship--It comes from the identity for the negative order associated Legendre functions.

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I don't agree, another phase, the so-called Racah phase, can be recognized by , which agrees with the usual definition of time-reversal. Racah multiplied Ylm with . Both phase conventions give easy to remember formulas for step up/down operations. Perhaps I should have stated: the CS convention can be recognized by this complex conjugation relation. P.wormer

"In quantum mechanically oriented texts the Condon and Shortley phase is usually introduced in the spherical harmonic functions. That is to say, the associated Legendre functions are defined without phase. The Wikipedia article defines the latter functions including the phase (-1)^m\,, which is why it is correctly absent in the definition of the spherical harmonic function in the present article."

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See Edmonds, Messiah (vol I), Brink and Satchler, Biedenharn and Louck (and I could mention more references) for Plm's without phase. My favorite definition of a spherical harmonic is one that does not need Plm's with negative m, because a Plm with negative m is somewhat unnatural and requires a new definition. See Altmann S. L.; Herzig P. Point-Group Theory Tables; Clarendon: Oxford, 1994. P.wormer

I think that we need to have a section devoted to the Condon-Shortley phase convention, because this becomes very confusing when discussing across disciplines. Some physicists put the phase within the Legendre function, whereas some append it the the definition of the spherical harmonic functions instead. Some disciplines (like geodesy and magnetics) do not use the CS phase at all. Honestly, I am not sure why people would want to use this phase, and the reaason for using it should be explained here. I think I heard that it simplifies operations involving ladder operators, but I am not sure.

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Yes ladder (=step up/down) operations become easier, they become the same for positive and negative m. Without CS (or Racah) phase one must distinguish the sign of m. P.wormer

Finally, we need to note if the analytic expressions for the Ylm's use the CS phase or not. Since I am guessing this section was written by a physicist, it probably does, be someone needs to check. Lunokhod 04:18, 6 November 2006 (UTC)

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The article as it is now is fine by me. P.wormer 13:15, 7 November 2006 (UTC)

[edit] Deleted images

I deleted this table of images. I find it very confusing, and am not even sure what cooridinate is being plotted. Lunokhod 18:41, 16 November 2006 (UTC)

Y10
Y20
Y30

No objection — between the little animation at the top and the graphic showing the partitioning of the spheres' surfaces by sign, I think one can get quite a good feel for these functions. There is that one graphic showing cross-sections of the sphere which generate the images you deleted; unless your images are brought back (which I would object to), I cannot see the purpose of this image either. I suggest deleting it too. Baccyak4H 18:58, 16 November 2006 (UTC)

Update. I removed the other image. Baccyak4H 19:00, 16 November 2006 (UTC)

[edit] To do: add identities

I think that this page is finally in decent shape. However, one thing that might be useful would be to have a list of commonly used identities, such as the integral of three harmonics expressed in wigner 3j coefficients, etc. I'll start this when I have the time, but this is low on my priority list. Is there anything else that is missing? Lunokhod 18:53, 16 November 2006 (UTC)

[edit] Real spherical harmonics

The transformation (given in this article) from complex to real spherical harmonics, will, if applied to Condon-Shortley spherical harmonics, give an m-dependent phase in the real harmonics. This is unpleasant. If this article were only meant for quantum mechanicians I would change it. But in another fields people have different opinions.--P.wormer 15:06, 6 July 2007 (UTC)

[edit] Laplace

I seem to have landed myself with the job of cleaning-up Pierre-Simon Laplace. The section Pierre-Simon Laplace#Spherical harmonics and potential theory needs attention from an expert. There seem to me to be two issues:

1. There is little point in explaining what spherical harmonics are, that is better done here; and
2. The definition (which if from Rouse Ball's (1908) A Short Account of the History of Mathematics) didn't immediately strike me as a suggestive definition.

Any thoughts on how to rewrite this section?Cutler 00:03, 25 August 2007 (UTC)

Rouse Ball gives the following account:

"This is memorable for the introduction into analysis of spherical harmonics or Laplace's coefficients, ...

"If the co-ordinates of two points be (r ,μ ,ω ) and (r′,μ' ,ω' ), and if r′ ≥ r, then the reciprocal of the distance between them can be expanded in powers of r/r′, and the respective coefficients are Laplace's coefficients. Their utility arises from the fact that every function of the co-ordinates of a point on the sphere can be expanded in a series of them. It should be stated that the similar coefficients for space of two dimensions, together with some of their properties, had been previously given by Adrien-Marie Legendre in a paper sent to the French Academy in 1783. Legendre had good reason to complain of the way in which he was treated in this matter."

Does anybody recognise this as a way of defining spherical harmonics? Should go in the article if so.Cutler 10:09, 25 August 2007 (UTC)